| L(s) = 1 | − 1.45e3·5-s − 2.18e3·9-s − 7.80e4·13-s − 1.31e5·17-s + 7.99e5·25-s − 4.04e5·29-s + 3.75e6·37-s + 6.18e6·41-s + 3.17e6·45-s + 2.19e6·49-s + 2.13e6·53-s − 3.43e7·61-s + 1.13e8·65-s − 1.06e8·73-s + 4.78e6·81-s + 1.91e8·85-s + 1.73e8·89-s − 1.47e8·97-s − 3.82e8·101-s − 1.37e8·109-s + 6.61e7·113-s + 1.70e8·117-s + 2.52e8·121-s + 1.70e8·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 2.32·5-s − 1/3·9-s − 2.73·13-s − 1.57·17-s + 2.04·25-s − 0.571·29-s + 2.00·37-s + 2.18·41-s + 0.774·45-s + 0.380·49-s + 0.270·53-s − 2.47·61-s + 6.35·65-s − 3.75·73-s + 1/9·81-s + 3.66·85-s + 2.76·89-s − 1.66·97-s − 3.67·101-s − 0.972·109-s + 0.405·113-s + 0.911·117-s + 1.17·121-s + 0.700·125-s + 1.32·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s+4)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.1385077789\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1385077789\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p^{7} T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 726 T + p^{8} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 914 p^{4} T^{2} + p^{16} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 252323090 T^{2} + p^{16} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 39034 T + p^{8} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 65814 T + p^{8} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 17000201234 T^{2} + p^{16} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 95455358590 T^{2} + p^{16} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 202062 T + p^{8} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 276239804882 T^{2} + p^{16} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 1876030 T + p^{8} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 3091050 T + p^{8} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 18251245763090 T^{2} + p^{16} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 7214640194114 T^{2} + p^{16} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 1066482 T + p^{8} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 260442349515410 T^{2} + p^{16} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 17154194 T + p^{8} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 59866697031314 T^{2} + p^{16} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 295210326091390 T^{2} + p^{16} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 53286014 T + p^{8} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 2700438986177234 T^{2} + p^{16} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 4443915113493650 T^{2} + p^{16} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 86667234 T + p^{8} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 73901822 T + p^{8} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.17513710424716372152590745335, −11.13402362399870035272037751365, −10.50132535522311251840591056381, −9.605679495620223711953035451261, −9.407756429300799375277238845743, −8.765986195443750263142854088507, −8.097004784722189091895479379890, −7.58594207577390372571239401753, −7.48489104635473035342332162386, −7.01299975256885672815583259933, −6.17092911418640255175017106012, −5.49429930017788776955338228201, −4.48370642596426067518237760931, −4.46925230223090905293057131511, −4.06683794083073691805518643085, −2.91902204570390844733044676306, −2.73517508324664461255625713971, −1.91507540596991583429384839084, −0.66558703464493010180718701347, −0.13454411129309862285288249384,
0.13454411129309862285288249384, 0.66558703464493010180718701347, 1.91507540596991583429384839084, 2.73517508324664461255625713971, 2.91902204570390844733044676306, 4.06683794083073691805518643085, 4.46925230223090905293057131511, 4.48370642596426067518237760931, 5.49429930017788776955338228201, 6.17092911418640255175017106012, 7.01299975256885672815583259933, 7.48489104635473035342332162386, 7.58594207577390372571239401753, 8.097004784722189091895479379890, 8.765986195443750263142854088507, 9.407756429300799375277238845743, 9.605679495620223711953035451261, 10.50132535522311251840591056381, 11.13402362399870035272037751365, 11.17513710424716372152590745335
